This paper generalizes our previous manuscript [1]} which established thm:main-informal for the low individual degree test. In addition we significantly simplify the earlier analysis of [1]} by working in the framework of synchronous strategies for the quantum provers. Synchronous strategies are a subclass of quantum strategies where the provers are assumed to use the same measurement operators whenever they receive the same question, and furthermore the entangled state that they share is a maximally entangled state. These assumptions allow us to shortcut many technical calculations that appear in [1]}. Furthermore, it is without loss of generality since recent work by one of us establishes that soundness with synchronous strategies for a large class of two-prover games (including the tensor code test) can be translated back into soundness for general quantum strategies [4]}. An additional benefit of our streamlined analysis is that synchronous strategies can also be defined in infinite dimensions. In this case the condition that the entangled state is maximally entangled is replaced by the condition that the state is tracial, a notion which we introduce formally below. Our proof naturally extends to this situation, and we are thus able to prove soundness of the tensor code test against infinite-dimensional (synchronous) quantum strategies. We explain the synchronous strategies framework in more detail in sec:synchronous. We motivate the consideration of infinite-dimensional strategies from a complexity-theoretic viewpoint in Section REF .
